LCM of 72 and 84
LCM of 72 and 84 is the smallest number among all common multiples of 72 and 84. The first few multiples of 72 and 84 are (72, 144, 216, 288, 360, 432, 504, . . . ) and (84, 168, 252, 336, 420, 504, . . . ) respectively. There are 3 commonly used methods to find LCM of 72 and 84  by listing multiples, by prime factorization, and by division method.
1.  LCM of 72 and 84 
2.  List of Methods 
3.  Solved Examples 
4.  FAQs 
What is the LCM of 72 and 84?
Answer: LCM of 72 and 84 is 504.
Explanation:
The LCM of two nonzero integers, x(72) and y(84), is the smallest positive integer m(504) that is divisible by both x(72) and y(84) without any remainder.
Methods to Find LCM of 72 and 84
Let's look at the different methods for finding the LCM of 72 and 84.
 By Division Method
 By Prime Factorization Method
 By Listing Multiples
LCM of 72 and 84 by Division Method
To calculate the LCM of 72 and 84 by the division method, we will divide the numbers(72, 84) by their prime factors (preferably common). The product of these divisors gives the LCM of 72 and 84.
 Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 72 and 84. Write this prime number(2) on the left of the given numbers(72 and 84), separated as per the ladder arrangement.
 Step 2: If any of the given numbers (72, 84) is a multiple of 2, divide it by 2 and write the quotient below it. Bring down any number that is not divisible by the prime number.
 Step 3: Continue the steps until only 1s are left in the last row.
The LCM of 72 and 84 is the product of all prime numbers on the left, i.e. LCM(72, 84) by division method = 2 × 2 × 2 × 3 × 3 × 7 = 504.
LCM of 72 and 84 by Prime Factorization
Prime factorization of 72 and 84 is (2 × 2 × 2 × 3 × 3) = 2^{3} × 3^{2} and (2 × 2 × 3 × 7) = 2^{2} × 3^{1} × 7^{1} respectively. LCM of 72 and 84 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2^{3} × 3^{2} × 7^{1} = 504.
Hence, the LCM of 72 and 84 by prime factorization is 504.
LCM of 72 and 84 by Listing Multiples
To calculate the LCM of 72 and 84 by listing out the common multiples, we can follow the given below steps:
 Step 1: List a few multiples of 72 (72, 144, 216, 288, 360, 432, 504, . . . ) and 84 (84, 168, 252, 336, 420, 504, . . . . )
 Step 2: The common multiples from the multiples of 72 and 84 are 504, 1008, . . .
 Step 3: The smallest common multiple of 72 and 84 is 504.
∴ The least common multiple of 72 and 84 = 504.
☛ Also Check:
 LCM of 15 and 18  90
 LCM of 13 and 14  182
 LCM of 10 and 45  90
 LCM of 4 and 15  60
 LCM of 8, 10 and 12  120
 LCM of 18 and 40  360
 LCM of 24, 30 and 40  120
LCM of 72 and 84 Examples

Example 1: Find the smallest number that is divisible by 72 and 84 exactly.
Solution:
The smallest number that is divisible by 72 and 84 exactly is their LCM.
⇒ Multiples of 72 and 84: Multiples of 72 = 72, 144, 216, 288, 360, 432, 504, . . . .
 Multiples of 84 = 84, 168, 252, 336, 420, 504, 588, . . . .
Therefore, the LCM of 72 and 84 is 504.

Example 2: The product of two numbers is 6048. If their GCD is 12, what is their LCM?
Solution:
Given: GCD = 12
product of numbers = 6048
∵ LCM × GCD = product of numbers
⇒ LCM = Product/GCD = 6048/12
Therefore, the LCM is 504.
The probable combination for the given case is LCM(72, 84) = 504. 
Example 3: Verify the relationship between GCF and LCM of 72 and 84.
Solution:
The relation between GCF and LCM of 72 and 84 is given as,
LCM(72, 84) × GCF(72, 84) = Product of 72, 84
Prime factorization of 72 and 84 is given as, 72 = (2 × 2 × 2 × 3 × 3) = 2^{3} × 3^{2} and 84 = (2 × 2 × 3 × 7) = 2^{2} × 3^{1} × 7^{1}
LCM(72, 84) = 504
GCF(72, 84) = 12
LHS = LCM(72, 84) × GCF(72, 84) = 504 × 12 = 6048
RHS = Product of 72, 84 = 72 × 84 = 6048
⇒ LHS = RHS = 6048
Hence, verified.
FAQs on LCM of 72 and 84
What is the LCM of 72 and 84?
The LCM of 72 and 84 is 504. To find the LCM of 72 and 84, we need to find the multiples of 72 and 84 (multiples of 72 = 72, 144, 216, 288 . . . . 504; multiples of 84 = 84, 168, 252, 336 . . . . 504) and choose the smallest multiple that is exactly divisible by 72 and 84, i.e., 504.
Which of the following is the LCM of 72 and 84? 504, 50, 45, 3
The value of LCM of 72, 84 is the smallest common multiple of 72 and 84. The number satisfying the given condition is 504.
If the LCM of 84 and 72 is 504, Find its GCF.
LCM(84, 72) × GCF(84, 72) = 84 × 72
Since the LCM of 84 and 72 = 504
⇒ 504 × GCF(84, 72) = 6048
Therefore, the greatest common factor (GCF) = 6048/504 = 12.
What is the Least Perfect Square Divisible by 72 and 84?
The least number divisible by 72 and 84 = LCM(72, 84)
LCM of 72 and 84 = 2 × 2 × 2 × 3 × 3 × 7 [Incomplete pair(s): 2, 7]
⇒ Least perfect square divisible by each 72 and 84 = LCM(72, 84) × 2 × 7 = 7056 [Square root of 7056 = √7056 = ±84]
Therefore, 7056 is the required number.
What is the Relation Between GCF and LCM of 72, 84?
The following equation can be used to express the relation between GCF and LCM of 72 and 84, i.e. GCF × LCM = 72 × 84.
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